Is it possible to set the search domain to be the nodes domain so that all containers become subdomains?

I want all my LXCs to be their own subdomain of the node they're on, which is node-1.lan.
I tried setting the search domain to be ".node-1.lan" instead of just ".lan" (it picked it up by default from my DHCP server) but it doesn't work, they're all still just <container>.lan.

Is this possible? I also tried manually setting the hostname of a container to container1.node-1, hoping for container1.node-1.lan, but it still just becomes container1.lan